Método da série de potências para solução de equações diferenciais ordinárias em torno de pontos ordinários

Abstract

Differential equations are part of a very significant theory in Mathematics, both due to their intrinsic nature, which encompasses a wide range of concepts, and their extensive applicability in different areas of knowledge. Although there are several systematic methods for determining solutions to differential equations, we still face considerable challenges in solving certain specific types of them. Thus, the objective of this study is to introduce the Power Serie Method for Solutions of Ordinary Differential Equations around Ordinary Points, which is not common in initial courses in Differential and Integral Calculus. This was carried out through bibliographical research, in which the main sources used in undergraduate Differential Equations courses were consulted. Initially, fundamental concepts related to differential equations were explored, including the discussion on the existence and uniqueness of solutions, as well as obtaining the general solution for linear differential equations, both homogeneous and non-homogeneous. Furthermore, the essential theory of numerical series and power series was revised to continue this work. Subsequently, the Power Serie Method for Solving Ordinary Differential Equations Around Ordinary Points was presented and finally we applied the method to obtain the solution of Legendre’s Equation. From the development of this work, it can be concluded that, even though there is a guarantee of the existence of solutions for a given differential equation, it is not always possible to express it in terms of elementary functions, that is, polynomial, rational, trigonometric functions, exponential and logarithmic. This highlights the importance of studying the power serie method.